Classes of piecewise-quasiaffine transformations on the generalized 2-group of quaternions

The class of nonabelian 2-groups $H$ with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group $H$, and put forward criteria of their bijectivity. For the generalized group of quaternions of order $2^m$, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.